## Teacher package: Geometry

Submitted by plusadmin on March 1, 2009

The *Plus* teacher packages are designed to give
teachers (and students) easy access to *Plus* content on a
particular subject area. Most *Plus* articles go far
beyond the explicit maths taught at school, while still being
accessible to someone doing A level maths. They put classroom
maths in context by explaining the bigger picture — they explore
applications in the real world, find maths in unusual places, and
delve into mathematical history and philosophy. We therefore hope
that our teacher packages provide an ideal resource for students
working on projects and teachers wanting to offer their students
a deeper insight into the world of maths.

## Geometry

This teacher package brings together all *Plus*
articles on geometry. There are plenty to choose from, so we have
divided them up into the categories below. Articles on a blue
background lend themselves particularly well for use in the
classroom because they contain explicit maths, or suggest easy to
do hands-on activities. The other articles paint the bigger
picture, provide intuitive insight, and give cultural
perspectives.

- Euclidean geometry in the plane
- Euclidean geometry in three dimensions and higher
- Fractal geometry
- Topology
- Non-Euclidean geometry
- Geometry in history, culture and art

Don't forget that our sister site NRICH has many hands-on problems, activities and articles covering geometry.

### Euclidean geometry in the plane

What is the area of a circle? — Several fun ways of working out the area of a circle.

Mathematical mysteries: trisecting the angle — An easy introduction — and resolution — of this ancient mystery.

Outer space: on a clear day — Just how far away is the horizon?

Mathematical mysteries: trisecting the angle — An easy introduction — and resolution — of this ancient mystery.

1089 and all that — Get your hands dirty with some surprising, but accessible, geometric results.

The trouble
with five — This article looks at the *five-fold tiling
problem*, which asks whether it's possible to tile the
plane with shapes that have a five-fold symmetry. It's a
step-by-step guide on how one would approach such a problem,
and it's right at the cutting edge of maths because it is still
unsolved. It's one of our most popular articles of all
time.

Secrets from a bathroom floor — Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. But while it sometimes seems free from worldly limitations, tiling is a very precise art, where not much can be left to chance. We can push and turn and wiggle, but if the maths is not right, it isn't going to tile.

From Kleenex to quasicrystals — This articles also investigates tilings of the plane in a hands-on way, and also takes us through some applications.

Catching primes — Primes are often caught in sieves, that of Eratosthenes being the most famous one. This article introduces a geometric algorithm for finding primes, and proves that it works using plane geometry.

Curious quaternions — This article explicitly describes the connection between complex numbers and plane geometry.

Number
crunching ants — This article explores *Buffon's
needle*, which is a way of approximating the value of pi by
throwing a needle on a sheet of paper, an activity that's
easily replicated. The article also explains how a species of
ants uses this algorithm to estimate the area of potential
nesting sites.

Can triangles help spot a bomb? — Airport security staff have a daunting task. With impatient queues looming over them they need to search x-ray scans of cluttered suitcases for several items at once: knives, guns and bombs. How can we ease their task and make sure they don't miss a crucial item? To find out, scientists are trying to understand how we humans take in visual information. The humble triangle plays a crucial role in the experiments they perform.

Bridges, string art and Bézier curves — The Jerusalem Chords Bridge, Israel, was built to make way for the city's light rail train system. Its design took into consideration more than just utility — it is a work of art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross cables, but also in the mathematics that lies behind it.

### Euclidean geometry in three dimensions and higher

Maths behind the rainbow — Rainbows: Keats complained that a mathematical explanation of these marvels of nature robs them of their magic, conquering "all mysteries by rule and line". But rainbow geometry is just as elegant as the rainbows themselves.

Conic section hide and seek — On a rainy night last month, in an ancient hall down a hidden alleyway in the centre of London, Bernard Silverman, Chief Scientific Advisor to the Home Office, revealed a surprising secret... ancient mathematics is at the heart of a very modern game of hide and seek.

Getting into the picture — This article explores the geometry of perspective and how it is used to produce three-dimensional reconstructions of the scenes depicted in medieval works of art.

Imaging maths — Unfolding polyhedra — This article explores polyhedra and how they can be unfolded into two-dimensional nets. It contains plenty of animations and images to guide intuition.

Maths goes to the movies — Three-dimensional vector geometry is essential in computer generated movies. This article gives some explicit examples of how it is used, and also looks at complex numbers and quaternions.

Analemmatic sundials: How to build one and why they work — Instructions for building a sundial and a geometric explanation of how it works.

Drinking coffee in the Klein Café — A quirky article about projections from three into two dimensions. It's a great exposition of the process of mathematical discovery.

Euler's polyhedron formula — Take any polyhedron, add the number of vertices and the number of faces, and then subtract the number of edges. The answer, for any polyhedron that doesn't have holes, is always 2! This article introduces this curious relationship, and shows how it can be used to classify the Platonic solids.

Exotic spheres, or why 4-dimensional space is a crazy place — The world we live in is strictly 3-dimensional. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like?

Meet the gyroid — What do butterflies, ketchup, microcellular structures, and plastics have in common? It's a curious minimal surface called the gyroid.

Shattering crystal symmetries — In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?

How long is a day? — The obvious answer is 24 hours, but in fact, the length of a day varies throughout the year. If you plot the position of the Sun in the sky at the same time every day, you get a strange figure of eight which has provided one artist with a source for inspiration.

Pylon of the month — Electricity pylons and why they're made out of triangles.

Swimming
in mathematics — The geometric structures behind the
*Watercube*, a venue in the 2008 Beijing Olympic
Games.

Kelvin's bubble
burst again — The geometric structures behind the
*Watercube*, a venue in the 2008 Beijing Olympic
Games.

Mathematical mysteries: Kepler's conjecture — This article explores one of great open problems in geometry: how to stack oranges.

Virtually reducing the 3D load — A news item reporting on an advance in computer generated movie techniques, involving three-dimensional "wire-meshes" of characters like Shrek.

They never saw it coming — Many predatory beasts use motion camouflaging to approach their prey undetected. This article looks at the vector geometry behind the phenomenon.

A symmetry approach to viruses — A look at the geometric symmetries that allow viruses to do their evil deeds.

Clever coiling — Something about nature loves a helix. This article explains why.

Thinking outside the box — This article explores the fourth dimension, finding the tesseract and more.

### Fractal geometry

How big is the Milky Way? — An explicit introduction to fractal geometry, with the mission of calculating the size of the Milky Way.

Measure for measure — This article explores measurable and unmeasurable sets, and introduces Cantor sets.

Extracting
beauty from chaos — This article shows how chaos can arise
from iterated functions, and introduces the *lyaponov
exponent*, a quantity that's behind beautiful fractal
images.

Modelling nature with fractals — How to create beautiful fractal landscapes and calculate their areas.

Outer space: Superficiality — How nature maximises — and minimises — surface area, and why.

Jackson's fractals — Another introduction to fractal dimension, motivated by the fractal structures in Jackson Pollock's drip paintings.

The artist's fractal fingerprint — More on the fractals structures in Pollock paintings.

Benoît Mandelbrot has died — This story reports on Mandelbrot's death and summarises his work, including the famous set that was named after him.

Pandora's 3D box — This story reports on a new 3D version of the Mandelbrot set.

### Topology

Imaging maths — Inside the Klein bottle — An introduction to the Klein bottle with plenty of animations and images.

Dividing walls: Topology and topography part I and part II — These two articles explore how topology can overcome topography and present some proofs along the way.

Hidden dimensions — That geometry should be relevant to physics is no surprise: after all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be.

A fly walks round a football — What makes a perfect football? Anyone who plays or simply watches the game could quickly list the qualities. The ball must be round, retain its shape, be bouncy but not too lively and, most importantly, be capable of impressive speeds. We find out that this last point is all down to the ball's surface, the most prized research goal in ball design.

Rubber data — Data, data, data — 21st century life provides tons of it. It's paradise for researchers, or at least it would be if we knew how to make sense of it all. But there's one method for doing this that's based on the kind of idea that gave us the London tube map.

Magnetic tangles — What happens when magnetic fields, like that of the Sun, get tangled up in knots?

Möbius at rest — A news item reporting on a mathematical breakthrough concerning the Möbius strip.

The Poincaré conjecture — The three articles Mathematical millionaire, Proof for Poincaré?, and The Fields Medals 2006 track the exciting recent history of the Poincaré conjecture.

Knitting by numbers — How to knit a Möbius strip and other tricky surfaces.

### Non-Euclidean geometry

How to
make a perfect plane — Two lines in a plane always intersect
in a single point ... *unless* the lines are parallel. This
annoying exception is constantly inserting itself into otherwise
simple mathematical statements. This article explains how to get
around the problem, introducing the projective
plane.

Mathematical mysteries: strange geometries — An easy introduction to non-Euclidean geometries via Euclid's fifth postulate.

Non-Euclidean geometry and Indra's pearls — An introduction to hyperbolic geometry and how it gives rise to beautiful fractal images. Comes with beautiful illustrations and movies.

### Geometry in history, culture and art

The power of origami — A look at origami and why it can solve to ancient geometry problems.

Innate geometry — An article exploring the evolution of geometric ability in humans.

We must know, we will know — The philosophy behind Euclid's geometry and how it influences modern mathematics.

Perfect buildings: The maths of modern architecture — An interview with members of the specialist modelling group at Foster + partners, who are behind building such as the Gherkin and the London Town Hall.

The art of numbers — A report on a maths and art project involving beautiful geometric shapes.

Symmetry rules — This articles shows how the concept of symmetry is not just relevant to geometry, and enables us to understand the secrets of our Universe.

Don't forget that our sister site NRICH has many hands-on problems, activities and articles covering geometry.